STANDARD 4.3: Mathematics

Standard 4.3 Patterns and Algebra All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes.
Big Idea Algebra provides language through which we communicate the patterns in mathematics.
4.3.5 A. Patterns
Descriptive Statement: Algebra provides the language through which we communicate the patterns in mathematics. From the earliest age, students should be encouraged to investigate the patterns that they find in numbers, shapes, and expressions, and by doing so, to make mathematical discoveries. They should have opportunities to analyze, extend, and create a variety of patterns and to use pattern-based thinking to understand and represent mathematical and other real-world phenomena.
Essential Questions	Enduring Understandings
- How can change be best represented mathematically? (4.5C1; 4.5F1; 4.5F2; 4.5F3; 4.5F4) - How can patterns, relations, and functions be used as tools to best describe and help explain real-life situations? (4.5C1)	- The symbolic language of algebra is used to communicate and generalize the patterns in mathematics. - Algebraic representation can be used to generalize patterns and relationships.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
1. Recognize, describe, extend, and create patterns involving whole numbers. · Descriptions using tables, verbal rules, simple equations, and graphs	Sample MC Item: Last year, the cafeteria at Kyle's school recycled 100 pounds of the trash that was collected. This year was the second year of recycling, and the cafeteria recycled twice as much. If the amount of trash the cafeteria recycles doubles each year, how much will be recycled in the fourth year? a. 1600 pounds * b. 800 pounds c. 600 pounds d. 400 pounds
4.3.5 B. Functions and Relationships
Descriptive Statement: The function concept is one of the most fundamental unifying ideas of modern mathematics. Student begin their study of functions in the primary grades, as they observe and study patterns. As students grow and their ability to abstract matures, students form rules, display information in a table or chart, and write equations which express the relationships they have observed. In high school, they use the more formal language of algebra to describe these relationships.
Essential Questions	Enduring Understandings
- How are patterns of change related to the behavior of functions? (4.5F1; 4.5F2; 4.5F3; 4.5F4)	- Patterns and relationships can be represented graphically, numerically, symbolically, or verbally. (4.5E1)
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
1. Describe arithmetic operations as functions, including combining operations and reversing them.
2. Graph points satisfying a function from T-charts, from verbal rules, and from simple equations
4.3.5 C. Modeling
Descriptive Statement: The function concept is one of the most fundamental unifying ideas of modern mathematics. Student begin their study of functions in the primary grades, as they observe and study patterns. As students grow and their ability to abstract matures, students form rules, display information in a table or chart, and write equations which express the relationships they have observed. In high school, they use the more formal language of algebra to describe these relationships.
Essential Questions	Enduring Understandings
- How are mathematical models used to describe physical relationships? (4.5E2) - How are physical models used to clarify mathematical relationships? (4.5E3)	- Mathematical models can be used to describe and quantify physical relationships. (4.5E2) - Physical models can be used to clarify mathematical relationships. (4.5E3)
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
1. Use number sentences to model situations. · Using variables to represent unknown quantities · Using concrete materials, tables, graphs, verbal rules, algebraic expressions/equations
2. Draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events. · Changes over time · Rates of change (e.g., when is plant growing slowly/rapidly, when is temperature dropping most rapidly/slowly)	Assessment Focus: • Students are asked to draw a graphical representation of a story.
4.3.5 D. Procedures
Descriptive Statement: Techniques for manipulating algebraic expressions - procedures - remain important, especially for students who may continue their study of mathematics in a calculus program. Utilization of algebraic procedures includes understanding and applying properties of numbers and operations, using symbols and variables appropriately, working with expressions, equations, and inequalities, and solving equations and inequalities.
Essential Questions	Enduring Understandings
- What makes an algebraic algorithm both effective and efficient? (4.5D1)	- Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole. - Reasoning and/or proof can be used to verify or refute conjectures or theorems in algebra. (4.5D1; 4.5D3; 4.5D4; 4.5D5)
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
1. Solve simple linear equations with manipulatives and informally · Whole-number coefficients only, answers also whole numbers · Variables on one side of equation